Nonlinear Dynamics

, Volume 95, Issue 3, pp 1903–1921 | Cite as

Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

  • Chaofeng LiEmail author
  • Peiyong Li
  • Bingfu Zhong
  • Bangchun Wen
Original Paper


The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.


Non-continuous elastic boundary Geometrically nonlinear Laminated cylindrical shells 

List of symbols

\({{\varvec{A}}}_{ij} \)

The stretching stiffness coefficients

\({\varvec{B}}_{ij} \)

The coupling stiffness coefficients


Damping matrix

\({\varvec{D}}_{ij} \)

The bending stiffness coefficients


Yong’s modulus in the principal directions


The error of the non-dimension y-coordinate of \(\hbox {NT}=i\) with respect to \(\hbox {NT}=6\)


The point harmonic excitation

\(G_{12} \)

Moduli of rigidity


Thickness of the shell

\({\varvec{K}}, {\varvec{K}}_{\mathrm{spr}}\)

Stiffness matrix, spring stiffness matrix


Length of the shell


Mass matrix

\({\varvec{M}}_{x},{\varvec{M}}_{\theta },{\varvec{M}}_{x\theta }\)

The moments of the in-plane stresses


The number of terms for circumferential wave

\({\varvec{N}}_x,{\varvec{N}}_\theta ,{\varvec{N}}_{x\theta }\)

The force of the in-plane stresses


The number of supported points


The number of supported arcs


The number of terms for Chebyshev polynomials


Plane stresses–strain matrix

\({\overline{{\varvec{Q}}}} \)

Transformation stiffness matrix

\({\varvec{Q}}^{\prime }\)

The nonlinear vector


Radius of the shell

\({\varvec{T}}_{\mathrm{s}} \)

Transformation matrix


Kinetic energy

\({\varvec{T}}_m^*\left( \xi \right) \)

The admissible displacement functions

\(U_\varepsilon , U_{\mathrm{spr}} \)

Strain energy, potential energy

\(\bar{{{\varvec{U}}}}, \bar{{{\varvec{V}}}}, \bar{{{\varvec{W}}}}\)

The mode vector satisfying a boundary condition

\(a_m , b_m , c_m \)

The unknown corresponding coefficients

\(f_0 \)

The amplitude of harmonic excitation

\(k_{u} ,k_{v} ,k_{w} ,k_{\theta }\)

Stiffness of axial, circumferential, radial, rotational spring per unit arc length

\({k}^{\prime }_u ,{k}^{\prime }_v ,{k}^{\prime }_w ,{k}^{\prime }_\theta \)

Stiffness of axial, circumferential, radial, rotational spring


The circumferential wave number


The generalized coordinates




Displacement in the x, \(\theta \), z directions

\(\alpha \)

The total length of supported arcs

\(\beta \)

Angular orientation of fibers

\(\varepsilon _x,\varepsilon _\theta ,\gamma _{x\theta }\)

The strains of the shell

\(\theta \)

The constraint radian

\(\theta _s,\theta '_s\)

The starting and ending radian of the sth arcs

\(\kappa _x,\kappa _\theta ,\kappa _{x\theta }\)

The curvature of the shell

\(\mu _{12},\mu _{21}\)

Poisson’s ratios

\(\xi \)

The non-dimensional axial coordinate

\(\rho \)

Mass density

\(\sigma _{\mathrm{x}} ,\sigma _{\mathrm{y}} ,\tau _{\mathrm{xy}}\)

The stresses of the shell

\(\omega \)

The frequency of harmonic excitation

\(\omega ^{*}\)

The non-dimensional natural frequency

\(\omega _d\)

The natural frequency of the shell



The Project is supported by the China Natural Science Funds (No. 51575093) and the Fundamental Research Funds for the Central Universities (Nos. N160313001 and N170308028).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mechanical Engineering and AutomationNortheastern UniversityShenyangChina
  2. 2.Key Laboratory of Vibration and Control of Aero-Propulsion Systems Ministry of Education of ChinaNortheastern UniversityShenyangChina

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